Yvonne,
Scotlandspeople has what looks to be your family in 1901 in Camlachie registration district, Lanarkshire
Owen 33, Mary 28, John 6, James 3, Thomas & Mary both 0 (twins presumably). Surname is spelled McCaffery.
1911 census has them in the same district. (They were living in Gt Eastern Rd, Glasgow). Owen 42, Mary 38, John 16, James 13, Mary 10, Thomas 10, Catherine 6, Owen 3. I child had died. Frank McAdam cousin 40 & Joseph Haughey 35 cousin, both single, were lodging with them.
Likely death of Mary Jane McCaffery in Glasgow 1971 aged 99. Her death certificate should hopefully confirm both her parents names.
Here’s a link to Owen & Mary’s marriage:
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1893/10615/5867336.pdf
Owen was a labourer from Tattykeel. His father John was a farmer and still alive. Mary’s townland looks to be Cargnary, though I can’t identify that. Corranarry is the nearest spelling I can find. Her father was Patrick Muldoon a labourer. Labourers moved around a lot to follow the available work which can make them hard to trace.
This may be Owen’s birth in 1868 at Tullybrian. Mother Phoebe Hughes.
https://civilrecords.irishgenealogy.ie/churchrecords/images/birth_returns/births_1868/03418/2253793.pdf
It’d be wise to cross reference that with his death certificate if you have it, to make sure it’s the right family. Scottish death certificates record both parents names.